Answer:
(1) 20,000 units should be produced in order to minimize the average cost per unit.
(2) The minimum average cost per unit is $700 per unit.
(3) The total cost of production at this level of output is $14,000,000.
Step-by-step explanation:
The given total cost function is correctly stated as follows:
C = 4,000,000 + 300q + 0.01q^2 …………………………… (1)
(1) How many units should be produced in order to minimize the average cost per unit?
AC = Average cost per unit = C / q
Substituting for C from equation (1), we have:
AC = (4,000,000 + 300q + 0.01q^2) / q …………………. (2)
Marginal cost can be obtained by taking the derivative of equation (1) as follows:
MC = C’ = 300 + (2 * 0.01)q
MC = 300 + 0.02q …………………………………………. (3)
AC is minimum when MC = AC. Therefore, equate equations (2) and (3) and solve for q as follows:
300 + 0.02q = (4,000,000 + 300q + 0.01q^2) / q
(300 + 0.02q)q = 4,000,000 + 300q + 0.01q^2
300q + 0.02q^2 = 4,000,000 + 300q + 0.01q^2
300q + 0.02q^2 - 300q - 0.01q^2 = 4,000,000
0.01q^2 = 4,000,000
q^2 = 4,000,000 / 0.01
q^2 = 400,000,000
q = 400,000,000^(1/2)
q = 20,000 units
Therefore, 20,000 units should be produced in order to minimize the average cost per unit.
(2) What is the minimum average cost per unit?
Substituting q = 20,000 into equation (2), we have:
AC = (4,000,000 + (300 * 20,000) + (0.01 * 20,000^2)) / 20,000
AC = $700 per unit
Therefore, the minimum average cost per unit is $700 per unit.
(3) What is the total cost of production at this level of output?
Substituting q = 20,000 into equation (1), we have:
C = 4,000,000 + (300 * 20,000) + (0.01 * 20,000^2)
C = $14,000,000
Therefore, the total cost of production at this level of output is $14,000,000.